Exercie 9) Ue Laplace ranform o wrie down he oluion o 2 x x = F in x = x x = v. wha phenomena do oluion o hi DE illurae (even hough we're forcing wih in co )? How would you have ried o olve hi problem in Chaper 3? raher han Exercie ) Solve he following IVP. Ue hi example o recall he general parial fracion algorihm. x 4 x = 8 e 2 x = x =
Wolfram check:
Mah 228- Week 3 April -4 Mon Apr 7.4-7.5 The following Laplace ranform maerial i ueful in yem where we urn forcing funcion on and off, and when we have righ hand ide "forcing funcion" ha are more complicaed han wha undeermined coefficien can handle. f wih f Ce M F f e d for M commen u a uni ep funcion e a for urning componen on and off a = a. f a u a e a F more complicaed on/off a e a uni impule/dela "funcion" f g d F G convoluion inegral o inver Laplace ranform produc The uni ep funcion wih jump a = i defined o be, u =,. IThi funcion i alo called he "Heaviide" funcion, e.g. in Maple and Wolfram alpha. In Wolfram alpha i' alo called he "hea" funcion. Oliver Heaviide wa a an accomplihed phyici in he 8'. The name i no becaue he graph i heavy on one ide. :-) hp://en.wikipedia.org/wiki/oliver_heaviide wih plo : plo Heaviide, = 3..3, color = green, ile = `graph of uni ep funcion` ; graph of uni ep funcion 3 2 2 3 Noice ha echnically he verical line hould no be here - a more precie picure would have a olid poin a, and a hollow circle a,, for he graph of u. In erm of Laplace ranform inegral definiion i doen' acually maer wha we define u o be.
Then, a ; i.e. a u a =, a ; i.e. a and ha graph ha i a horizonal ranlaion by a o he righ, of he original graph, e.g. for a = 2: Exercie ) Verify he able enrie u a uni ep funcion e a for urning componen on and off a = a. f a u a e a F more complicaed on/off
Exercie 2) Conider he funcion f which i zero for 4 and wih he following graph. Ue lineariy and he uni ep funcion enry o compue he Laplace ranform F. Thi hould remind you of a homework problem from he aignmen due omorrow - alhough you're aked o find he Laplace ranform of ha ep funcion direcly from he definiion. In your nex week' homework aignmen you will re-do ha problem uing uni ep funcion. (Of coure, you could alo check your anwer in hi week' homework wih hi mehod.) 2 2 3 4 5 6 7 8
Exercie 3a) Explain why he decripion above lead o he differenial equaion iniial value problem for x x x =.2 co u x = x = 3b) Find x. Show ha afer he paren op puhing, he child i ocillaing wih an ampliude of exacly meer (in our linearized model).
Picure for he wing: plo plo. in, =.. Pi, color = black : plo2 plo Pi in, = Pi..2 Pi, color = black : plo3 plo Pi, = Pi..2 Pi, color = black, lineyle = 2 : plo4 plo Pi, = Pi..2 Pi, color = black, lineyle = 2 : plo5 plo., =.. Pi, color = black, lineyle = 2 : plo6 plo., =.. Pi, color = black, lineyle = 2 : diplay plo, plo2, plo3, plo4, plo5, plo6, ile = `advenure a he winge` ; 3 3 advenure a he winge 2 4 6 8 2 6 2 Alernae approach via Chaper 3: ep ) olve for. ep 2) Then olve and e x = y for. x x =.2 co x = x = y y = y = x y = x
f, wih f Ce M F f e d for M verified c f c 2 f 2 c F c 2 F 2 2 n, n 2 2 3 n! n ( e ( a co k in k coh k inh k e a co k e a in k e a f u a f a u a a f f f n, n 2 k 2 ( k 2 k 2 ( 2 k 2 ( k k 2 k 2 ( k a a 2 k 2 ( a k a 2 k 2 a F a e a e a F e a F f 2 F f f n F n f... f n
f d F f 2 f n f, n f co k 2 k in k 2 k 3 in k k co k e a n e a, n F F n F n F d 2 k 2 2 k 2 2 2 k 2 2 2 k 2 2 a 2 n! a n f g d F G f wih period p e p p f e d Laplace ranform able
EP 7.6 impule funcion and he operaor. Conider a force f acing on an objec for only on a very hor ime inerval a a, for example a when a ba hi a ball. Thi impule p of he force i defined o be he inegral p a a f d and i meaure he ne change in momenum of he objec ince by Newon' econd law m v = f a a m v d = m v a a a = a f d = p Since he impule p only depend on he inegral of f, and ince he exac form of f i unlikely o be known in any cae, he eaie model i o replace f wih a conan force having he ame oal impule, i.e. o e f = p d a, where d a, i he uni impule funcion given by = p., a, a a d a, =, a. Noice ha a d a, d = a d =. a a Here' a graph of d 2,., for example: 6 2 3 4 Since he uni impule funcion i a linear combinaion of uni ep funcion, we could olve differenial equaion wih impule funcion o-conruced. A far a Laplace ranform goe, i' even eaier o ake he limi a for he Laplace ranform d a,, and hi effecively model impule on very hor ime cale. d a, = u a u a
d a, = e a e a = e a e. In Laplace land we can ue L'Hopial' rule (in he variable ) o ake he limi a : lim e a e = e a e lim = e a. The reul in ime pace i no really a funcion bu we call i he "dela funcion" a anyway, and viualize i a a funcion ha i zero everywhere excep a = a, and ha i i infinie a = a in uch a way ha i inegral over any open inerval conaining a equal one. A explained in EP7.6, he dela "funcion" can be hough of in a rigorou way a a linear ranformaion, no a a funcion. I can alo be hough of a he derivaive of he uni ep funcion u a, and hi i conien wih he Laplace able enrie for derivaive of funcion. In any cae, hi lead o he very ueful Laplace ranform able enry a uni impule funcion e a for impule forcing
Exercie 4) Revii he wing. In hi cae he paren i providing an impule each ime he child pae hrough equilibrium poiion afer compleing a cycle. x x =.2 2 4 6 8 x = x =. wih plo : plo plo. in, =.. Pi, color = black : plo2 plo Pi in, = Pi..2 Pi, color = black : plo3 plo Pi, = Pi..2 Pi, color = black, lineyle = 2 : plo4 plo Pi, = Pi..2 Pi, color = black, lineyle = 2 : plo5 plo., =.. Pi, color = black, lineyle = 2 : plo6 plo., =.. Pi, color = black, lineyle = 2 : diplay plo, plo2, plo3, plo4, plo5, plo6, ile = `Wedneday advenure a he winge` ; 3 3 Wedneday advenure a he winge 2 4 6 8 2 4 6 8 2 impule oluion: five equal impule o ge ame final ampliude of meer - Exercie : f.2 Pi um Heaviide k 2 Pi in k 2 Pi, k =..4 : plo f, =..2 Pi, color = black, ile = `lazy paren on Friday` ; 3 3 Or, an impule a = and anoher one a =. lazy paren on Friday 2 4 6 8 4 2
g.2 Pi 2 in 3 Heaviide Pi in Pi : plo g, =..2 Pi, color = black, ile = `very lazy paren` ; 3 3 very lazy paren 2 4 6 8 4 2
Mah 228- Wed Apr 2 Convoluion and oluion o non-homogeneou phyical ocillaion problem (EP7.6 p. 499-5) Conider a mechanical or elecrical forced ocillaion problem for x, and he paricular oluion ha begin a re: a x b x c x = f x = x =. Then in Laplace land, hi equaion i equivalen o a 2 X b X c X = F X a 2 b c = F X = F a 2 b c F W. The invere Laplace ranform w = W i called he "weigh funcion" of he given differenial equaion. Noice (check!) ha w i he oluion o he homogeneou DE IVP a x cb x c x = x = x = Becaue of he convoluion able enry f g d F G convoluion inegral o inver Laplace ranform produc he oluion (for ANY forcing funcion f ) i given by x = f w d. (Wih non-zero iniial condiion here would be homogeneou oluion erm a well. In he cae of damping hee erm would be ranien.) Noice ha hi ay ha x depend on he value of he forcing funcion f for he previou ime, weighed by w,. Tha he non-homogenou oluion can be conruced from he homogeneou one via hi convoluion i a pecial cae of "Duhamel' Principle", which applie o linear DE' and linear PDE': hp://en.wikipedia.org/wiki/duhamel%27_principle Thi idea generalize o much more complicaed mechanical and circui yem, and i how engineer experimen mahemaically wih how propoed configuraion will repond o variou inpu forcing funcion, once hey figure ou he weigh funcion for heir yem. The mahemaical juificaion for he general convoluion able enry i a he end of oday' noe.
Exercie 2. Le' play he reonance game and pracice convoluion inegral, fir wih an old friend, bu hen wih non-inuoidal forcing funcion. We'll ick wih our earlier wing, bu conider variou forcing periodic funcion f. x x = f x = x = a) Find he weigh funcion w. b) Wrie down he oluion formula for x a a convoluion inegral. c) Work ou he pecial cae of X when f = co, and verify ha he convoluion formula reproduce he anwer we would've goen from he able enry 2 k in k 2 k 2 2 in co d ; co in d ; #convoluion i commuaive 2 in 2 in d) Then play he reonance game on he following page wih new periodic forcing funcion... ()
We worked ou ha he oluion o our DE IVP will be Example ) A quare wave forcing funcion wih ampliude and period 2. Le' alk abou how we came up wih he formula (which work unil = ). wih plo : x = in f d Since he unforced yem ha a naural angular frequency =, we expec reonance when he forcing funcion ha he correponding period of T = 2 = 2. We will dicover ha here i he poibiliy w for reonance if he period of f i a muliple of T. (Alo, forcing a he naural period doen' guaranee reonance...i depend wha funcion you force wih.) f 2 n = n Heaviide n Pi : ploa plo f, =..3, color = green : diplay ploa, ile = `quare wave forcing a naural period` ; quare wave forcing a naural period 2 3 ) Wha' your voe? I hi quare wave going o induce reonance, i.e. a repone wih linearly growing ampliude? x in f d : plob plo x, =..3, color = black : diplay ploa, plob, ile = `reonance repone?` ; reonance repone? 2 3
Example 2) A riangle wave forcing funcion, ame period f2 f d.5 : # hi aniderivaive of quare wave hould be riangle wave plo2a plo f2, =..3, color = green : diplay plo2a, ile = `riangle wave forcing a naural period` ; 2) Reonance? riangle wave forcing a naural period 2 3 x2 in f2 d : plo2b plo x2, =..3, color = black : diplay plo2a, plo2b, ile = `reonance repone?` ; reonance repone? 2 3
Example 3) Forcing no a he naural period, e.g. wih a quare wave having period T = 2. f3 2 2 n = n Heaviide n : plo3a plo f3, =..2, color = green : diplay plo3a, ile = `periodic forcing, no a he naural period` ; 3) Reonance? periodic forcing, no a he naural period 5 5 2 x3 in f3 d : plo3b plo x3, =..2, color = black : diplay plo3a, plo3b, ile = `reonance repone?` ; reonance repone? 5 5 2
Example 4) Forcing no a he naural period, e.g. wih a paricular wave having period T = 6. f4 n = Heaviide 6 n Heaviide 6 n : plo4a plo f4, =..5, color = green : diplay plo4a, ile = poradic quare wave wih period 6 ; poradic quare wave wih period 6 5 5 4) Reonance? x4 in f4 d : plo4b plo x4, =..5, color = black : diplay plo4a, plo4b, ile = `reonance repone?` ; reonance repone? 5 5
Hey, wha happened???? How do we need o modify our hinking if we force a yem wih omehing which i no inuoidal, in erm of worrying abou reonance? In he cae ha hi wa modeling a wing (pendulum), how i i geing puhed? Precie Anwer: I urn ou ha any periodic funcion wih period P i a (poibly infinie) uperpoiion P of a conan funcion wih coine and ine funcion of period P, 2, P 3, P,.... Equivalenly, hee 4 funcion in he uperpoiion are, co, in, co 2, in 2, co 3, in 3,... wih ω = 2. Thi i he P heory of Fourier erie, which you will udy in oher coure, e.g. Mah 35, Parial Differenial Equaion. If he given periodic forcing funcion f ha non-zero erm in hi uperpoiion for which n = (he naural angular frequency) (equivalenly P n = 2 = T ), here will be reonance; oherwie, no reonance. We could already have underood ome of hi in Chaper 3, for example Exercie 3) The naural period of he following DE i (ill) T = 2 forcing funcion below i T = 6. Noice ha he period of he fir and ha he period of he econd one i T = T = 2. Ye, i i he fir DE whoe oluion will exhibi reonance, no he econd one. Explain, uing Chaper 5 uperpoiion idea. a) x x = co in. 3 b) x x = co 2 3 in 3.
Mah 228- Fri Apr 4 Chaper 9 Fourier Serie and applicaion o differenial equaion (and parial differenial equaion) 9.-9.2 Fourier erie definiion and convergence. The idea of Fourier erie i relaed o he linear algebra concep of do produc, norm, and projecion. We'll review hi connecion afer he definiion of Fourier erie: Le f :, be a piecewie coninuou funcion, or equivalenly, exend o f : a a 2 periodic funcion. Example one could conider he 2 -periodic exenion of f =, iniially defined on he inerval,, o all of. I graph i he o-called "en funcion", en The Fourier coefficen of a 2 3 2 3 2 2 3 periodic funcion f are compued via he definiion a f d a n f co n d, n b n f in n d, n And he Fourier erie for f i given by f a a 2 n co n n = b n in n. n = The idea i ha he parial um of he Fourier erie of f hould acually converge o f. The reaon why hi hould be rue combine linear algebra idea relaed o orhonormal bai vecor and projecion, wih analyi idea relaed o convergence. Le' do an example o illurae he magic, before dicuing (par of) why he convergence acually happen.
Exercie Conider he even funcion f = on he inerval, exended o be he 2 periodic "en funcion" en of page. Find he Fourier coefficien a, a n, b n and he Fourier erie for en. The anwer i below, along wih a graph of parial um of he Fourier erie. 3 3 2 3 oluion: en 2 4 n odd n 2 co n f 2 4 4 j = 2 j 2 co 2 j : plo f, =.., color = black ; 3 2 5 5
Uing echnology o compue Fourier coefficien: f ; f := (2) a f d; aume n, ineger ; # hi will le Maple aemp o evaluae he inegral a n f co n d : b n f in n d : a n ; b n ; a := 2 n~ n~ 2 (3)
So wha' going on? Recall he idea of do produc, angle, orhonormal bai and projecion ono ubpace, in N, from linear algebra: For x, y N, he do produc x y a) x x and = if and only if x = b) x y = y x c) x y z = x y x z d) x y = x y = x y N k = x k y k aifie for all vecor x, y, z N and calar : From hee four properie one can define he norm or magniude of a vecor by x = x x and he diance beween wo vecor x, y by di x, y x y. One can check wih algebra ha he Cauchy-Schwarz inequaliy hold: x y x y, wih equaliy if and only if x, y are calar muliple of each oher. One conequence of he Cauchy- Schwarz inequaliy i he riangle inequaliy x y x y, wih equaliy if and only if x, y are non-negaive calar muliple of each oher. Equivalenly, in erm of Euclidean diance, di x, z di x, y di y, z. Anoher conequence of he Cauchy-Schwarz inequaliy i ha one can define he angle beween x, y via x y co, x y x y for, becaue hold o ha exi. In paricular wo vecor x, y are x y perpendicular, or orhogonal if and only if x y =. If one ha a n dimenional ubpace W N an orhonormal bai u, u 2,... u n for W i a collecion of uni vecor (normalized o lengh ), which are alo muually orhogonal. (One can find uch bae via he Gram-Schmid algorihm.) For uch an orho-normal bai he neare poin projecion of a vecor x N ono W i given by proj W x = x u u x u 2 u 2... x u n u n = For any x (already) in W, proj W x = x. n k = x u k u k.
The enire algebraic/geomeric developmen on he previou page ju depended on he four algebraic properie a,b,c,d for he do produc. So i can be generalized: Definiion Le V i any real-calar vecor pace. we call V an inner produc pace if here i an inner produc f, g for which he inner produc aifie f, g, h V and calar : a) f, f. f, f =. b) f, g = g, f. c) f, g h = f, g f, h d) f, g = f, g = f, g. In hi cae one can define f = f, f, di f, g = f g ; prove he Cauchy-Schwarz inequaliy and he riangle inequaliie; define angle beween vecor, and in paricular, orhogonaliy beween vecor; find orho-normal bae u, u 2,... u n for finie-dimenional ubpace W, and prove ha for any f V he neare elemen in W o f i given by proj W f = f, u u f, u 2 u 2... f, u n u n = n k = f, u k u k. Theorem Le V = f :.. f i piecewie coninuou and 2 periodic. Define f, g f g d. ) Then V,, i an inner produc pace. 2) Le V N pan, co, co 2,..., co N, in, in 2,... in N. Then he 2 2 N funcion lied in hi collecion are an orhonormal bai for he 2 N dimenional ubpace V N. In paricular, for any f V he neare funcion in V N o f i given by proj VN f = f, 2 N 2 n = N = a 2 n = f, co n co n a n co n where a, a n, b n are he Fourier coefficien defined on page. N n = b n in n N n = f, in n in n
Exercie 2) Check ha, co, co 2,..., co N, in, in 2,... in N 2 are orhonormal wih repec o he inner produc f, g f g d o Hin: co m k = co m co k in m in k in m k = in m co k co m in k co m co k = 2 co m k co m k (even if m = k in m in k = 2 co m k co m k (even if m = k co m in k = 2 in m k in m k
Exercie 3) Conider he 2 a a 2 periodic funcion. periodic odd funcion aw define by exending f =, awooh funcion 3 3 2 2 3 3 Find he Fourier erie for aw. Hin: you noiced ha for he even en funcion in Exercie he ine Fourier coefficien were all zero. Which one will be zero for any odd funcion? Why? oluion: aw 2 n = n n in n f2 2 n = n n in n : plo f2, =.., color = black ; 3 5 2 5
Convergence Theorem (Thee require ome careful mahemaical analyi o prove - hey are ofen dicued in Mah 52, for example.) Theorem Le f : be 2 periodic and piecewie coninuou. Le f N = proj VN f = a be he Fourier erie runcaed a N. Then lim n f f N N 2 n = = lim n a n co n f f N N n = b n in n In oher word, he diance beween f N and f converge o zero, where we are uing he diance funcion ha we ge from he inner produc, 2 d 2 =. 2 di f, g = f g = f g, f g = f g 2 d. Theorem 2 If f i a in Theorem, and i (alo) piecewie differeniable wih a mo jump diconinuiie, hen (i) for any uch ha f i differeniable a lim f N N = f (poinwie convergence). (ii) for any where f i no differeniable (bu i eiher coninuou or ha a jump diconinuiy), hen where lim N f N = 2 f f f = lim f, f = lim f Example: ) The runcaed Fourier erie for he en funcion, en N converge o en for all. In fac, i can be hown ha he convergence i uniform, i.e. N.. n N en en n for all a once. 2) The runcaed Fourier erie for he awooh funcion, aw N converge o aw for all 2 k, k (i.e. everywhere excep a he jump poin). A hee jump poin he Fourier erie converge o he average of he lef and righ hand limi of aw, which i. (In fac, each parial um evaluae o a hoe poin.) The convergence a he oher value i poinwie, bu no uniform, a he convergence ake longer nearer he jump poin.)
Exercie 4) We can derive "magic" ummaion formula uing Fourier erie. (See your homework for ome more.) From Theorem 2 we know ha he Fourier erie for en converge for all. In paricular 4a) Deduce 4b) Verify and ue o how = en = 2 4 n odd 3 2 5 2... = n odd n = n 2 = = n odd n 2 n odd n = n 2 co n. n even n 2 4 n = n 2 n 2 = 2 6. n 2 = n 2 2 8.
Differeniaing Fourier Serie: Theorem 3 Le f be 2 periodic, piecewie differeniable and coninuou, and wih f piecewie coninuou. Le f have Fourier erie a 2 n = f a n co n b n in n. n = Then f ha he Fourier erie you'd expec by differeniaing erm by erm: proof: Le f have Fourier erie f n = n a n in n n b n co n n = Then f A A 2 n co n n = B n in n. n = A n = f co n d, n. Inegrae by par wih u = co n, dv = f d, du = n in n d, v = f : f co n d = f n in n f n in n d = n f in n d = n b n. Similarly, A =, B n = n a n. Remark: Thi i analogou o wha happened wih Laplace ranform. In ha cae, he ranform of he derivaive muliplied he ranform of he original funcion by (and here were correcion erm for he iniial value). In hi cae he ranformed variable are he a n, b n which depend on n. And he Fourier erie "ranform" of he derivaive of a funcion muliplie hee coefficien by n (and permue hem).
Exercie 5a Ue he differeniaion heorem and he Fourier erie for en o find he Fourier erie for he quare wave, quare, which i he 2 periodic exenion of f = quare() 3 2 2 3 (You will find he erie direcly from he definiion in your homework.) 5b) Deduce he magic formula 3 5 7...= k = k 2 k = 4. oluion: quare 4 n odd n in n f3 4 n = 2 n in 2 n : plo f3, =.., color = black ; 5 5 Could you check he Fourier coefficien wih echnology?